# Continuous

Pronunciation: /kənˈtɪn.ju.əs/ Explain
 Click on the point on the black slider and drag it to see various functions. Click on the blue point on the x-axis and drag it to see the values of the functions for the values of x. All of the discontinuous functions shown have a discontinuity at x=1. Manipulative 1 - Discontinuous Functions Created with GeoGebra.

A function or curve is continuous if it is an unbroken curve. A more rigorous definition of continuity is:

A function is continuous at x if f(x) is defined and .

A function is continuous on a subdomain if it is continuous at all points in the subdomain. A function is continuous on the entire domain if it is continuous at all points in the domain. A function is discontinuous if it is not continuous.

There is a distinction between continuous on the left and continuous on the right. Take the ceiling function at x = 1 as an example. At x = 1, the ceiling function is continuous on the left, as the limit of f(a) as a approaches 1 from the left is 1. However, the limit of f(a) as a approaches 1 from the right is 2. To be continuous at x = 1, f(x) must be continuous from the left and continuous from the right.

Continuity is important in analyzing functions, particularly in calculus. In calculus, one finds the derivative of a function. The derivative of a function exists only on subdomains that are continuous.

### References

1. McAdams, David E.. All Math Words Dictionary, continuous. 2nd Classroom edition 20150108-4799968. pg 45. Life is a Story Problem LLC. January 8, 2015. Buy the book

McAdams, David E. Continuous. 4/16/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/c/continuous.html.

### Revision History

4/16/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)